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Arrowhead Matrix
In the mathematical field of linear algebra, an arrowhead matrix is a square matrix containing zeros in all entries except for the first row, first column, and main diagonal, these entries can be any number. In other words, the matrix has the form : A=\begin \,\! *&*&*&*&* \\ \,\! *&*&0&0&0 \\ \,\! *&0&*&0&0 \\ \,\! *&0&0&*&0 \\ \,\! *&0&0&0&* \end. Any symmetric permutation of the arrowhead matrix, P^T A P, where ''P'' is a permutation matrix, is a (permuted) arrowhead matrix. Real symmetric arrowhead matrices are used in some algorithms for finding of eigenvalues and eigenvectors. Real symmetric arrowhead matrices Let ''A'' be a real symmetric (permuted) arrowhead matrix of the form : A = \begin D & z \\ z^ & \alpha \end, where D=\mathop(d_,d_,\ldots ,d_) is diagonal matrix of order ''n''−1, z=\begin \zeta _ & \zeta _ & \cdots & \zeta _ \end^T is a vector and \alpha is a scalar. Note that here the arrow is pointing to the bottom right. Let : A=V\Lambda V^ be the eigenv ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Algebra
Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrices. Linear algebra is central to almost all areas of mathematics. For instance, linear algebra is fundamental in modern presentations of geometry, including for defining basic objects such as lines, planes and rotations. Also, functional analysis, a branch of mathematical analysis, may be viewed as the application of linear algebra to spaces of functions. Linear algebra is also used in most sciences and fields of engineering, because it allows modeling many natural phenomena, and computing efficiently with such models. For nonlinear systems, which cannot be modeled with linear algebra, it is often used for dealing with first-order approximations, using the fact that the differential of a multivariate function at a point is the linear ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square Matrix
In mathematics, a square matrix is a matrix with the same number of rows and columns. An ''n''-by-''n'' matrix is known as a square matrix of order Any two square matrices of the same order can be added and multiplied. Square matrices are often used to represent simple linear transformations, such as shearing or rotation. For example, if R is a square matrix representing a rotation (rotation matrix) and \mathbf is a column vector describing the position of a point in space, the product R\mathbf yields another column vector describing the position of that point after that rotation. If \mathbf is a row vector, the same transformation can be obtained using where R^ is the transpose of Main diagonal The entries a_ (''i'' = 1, …, ''n'') form the main diagonal of a square matrix. They lie on the imaginary line which runs from the top left corner to the bottom right corner of the matrix. For instance, the main diagonal of the 4×4 matrix above contains the elements , , , . The d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Permutation Matrix
In mathematics, particularly in matrix theory, a permutation matrix is a square binary matrix that has exactly one entry of 1 in each row and each column and 0s elsewhere. Each such matrix, say , represents a permutation of elements and, when used to multiply another matrix, say , results in permuting the rows (when pre-multiplying, to form ) or columns (when post-multiplying, to form ) of the matrix . Definition Given a permutation of ''m'' elements, :\pi : \lbrace 1, \ldots, m \rbrace \to \lbrace 1, \ldots, m \rbrace represented in two-line form by :\begin 1 & 2 & \cdots & m \\ \pi(1) & \pi(2) & \cdots & \pi(m) \end, there are two natural ways to associate the permutation with a permutation matrix; namely, starting with the ''m'' × ''m'' identity matrix, , either permute the columns or permute the rows, according to . Both methods of defining permutation matrices appear in the literature and the properties expressed in one representation can be easily converted to th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted by \lambda, is the factor by which the eigenvector is scaled. Geometrically, an eigenvector, corresponding to a real nonzero eigenvalue, points in a direction in which it is stretched by the transformation and the eigenvalue is the factor by which it is stretched. If the eigenvalue is negative, the direction is reversed. Loosely speaking, in a multidimensional vector space, the eigenvector is not rotated. Formal definition If is a linear transformation from a vector space over a field into itself and is a nonzero vector in , then is an eigenvector of if is a scalar multiple of . This can be written as T(\mathbf) = \lambda \mathbf, where is a scalar in , known as the eigenvalue, characteristic value, or characteristic root ass ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eigenvector
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted by \lambda, is the factor by which the eigenvector is scaled. Geometrically, an eigenvector, corresponding to a real nonzero eigenvalue, points in a direction in which it is stretched by the transformation and the eigenvalue is the factor by which it is stretched. If the eigenvalue is negative, the direction is reversed. Loosely speaking, in a multidimensional vector space, the eigenvector is not rotated. Formal definition If is a linear transformation from a vector space over a field into itself and is a nonzero vector in , then is an eigenvector of if is a scalar multiple of . This can be written as T(\mathbf) = \lambda \mathbf, where is a scalar in , known as the eigenvalue, characteristic value, or characteristic root ass ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eigendecomposition Of A Matrix
In linear algebra, eigendecomposition is the Matrix factorization, factorization of a matrix (mathematics), matrix into a canonical form, whereby the matrix is represented in terms of its eigenvalues and eigenvectors. Only diagonalizable matrix, diagonalizable matrices can be factorized in this way. When the matrix being factorized is a normal matrix, normal or real symmetric matrix, the decomposition is called "spectral decomposition", derived from the spectral theorem. Fundamental theory of matrix eigenvectors and eigenvalues A (nonzero) vector of dimension is an eigenvector of a square matrix if it satisfies a linear equation of the form :\mathbf \mathbf = \lambda \mathbf for some scalar . Then is called the eigenvalue corresponding to . Geometrically speaking, the eigenvectors of are the vectors that merely elongates or shrinks, and the amount that they elongate/shrink by is the eigenvalue. The above equation is called the eigenvalue equation or the eigenvalue probl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Standard Basis
In mathematics, the standard basis (also called natural basis or canonical basis) of a coordinate vector space (such as \mathbb^n or \mathbb^n) is the set of vectors whose components are all zero, except one that equals 1. For example, in the case of the Euclidean plane \mathbb^2 formed by the pairs of real numbers, the standard basis is formed by the vectors :\mathbf_x = (1,0),\quad \mathbf_y = (0,1). Similarly, the standard basis for the three-dimensional space \mathbb^3 is formed by vectors :\mathbf_x = (1,0,0),\quad \mathbf_y = (0,1,0),\quad \mathbf_z=(0,0,1). Here the vector e''x'' points in the ''x'' direction, the vector e''y'' points in the ''y'' direction, and the vector e''z'' points in the ''z'' direction. There are several common notations for standard-basis vectors, including , , , and . These vectors are sometimes written with a hat to emphasize their status as unit vectors (standard unit vectors). These vectors are a basis in the sense that any other vector can ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Min-max Theorem
In linear algebra and functional analysis, the min-max theorem, or variational theorem, or Courant–Fischer–Weyl min-max principle, is a result that gives a variational characterization of eigenvalues of compact Hermitian operators on Hilbert spaces. It can be viewed as the starting point of many results of similar nature. This article first discusses the finite-dimensional case and its applications before considering compact operators on infinite-dimensional Hilbert spaces. We will see that for compact operators, the proof of the main theorem uses essentially the same idea from the finite-dimensional argument. In the case that the operator is non-Hermitian, the theorem provides an equivalent characterization of the associated singular values. The min-max theorem can be extended to self-adjoint operators that are bounded below. Matrices Let be a Hermitian matrix. As with many other variational results on eigenvalues, one considers the Rayleigh–Ritz q ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Givens Rotation
In numerical linear algebra, a Givens rotation is a rotation in the plane spanned by two coordinates axes. Givens rotations are named after Wallace Givens, who introduced them to numerical analysts in the 1950s while he was working at Argonne National Laboratory. Matrix representation A Givens rotation is represented by a matrix of the form :G(i, j, \theta) = \begin 1 & \cdots & 0 & \cdots & 0 & \cdots & 0 \\ \vdots & \ddots & \vdots & & \vdots & & \vdots \\ 0 & \cdots & c & \cdots & -s & \cdots & 0 \\ \vdots & & \vdots & \ddots & \vdots & & \vdots \\ 0 & \cdots & s & \cdots & c & \cdots & 0 \\ \vdots & & \vdots & & \vdots & \ddots & \vdots \\ 0 & \cdots & 0 & \cdots & 0 & \cdots & 1 \end, where and appear at the int ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Molecular Electronic Transition
Molecular electronic transitions take place when electrons in a molecule are excited from one energy level to a higher energy level. The energy change associated with this transition provides information on the structure of a molecule and determines many molecular properties such as colour. The relationship between the energy involved in the electronic transition and the frequency of radiation is given by Planck's relation. Organic molecules and other molecules The electronic transitions in organic compounds and some other compounds can be determined by ultraviolet–visible spectroscopy, provided that transitions in the ultraviolet (UV) or visible range of the electromagnetic spectrum exist for this compound. Electrons occupying a HOMO of a sigma bond can get excited to the LUMO of that bond. This process is denoted as a σ → σ* transition. Likewise promotion of an electron from a π-bonding orbital to an antibonding π orbital* is denoted as a π → π* transition. Au ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fermi Liquid Theory
Fermi liquid theory (also known as Landau's Fermi-liquid theory) is a theoretical model of interacting fermions that describes the normal state of most metals at sufficiently low temperatures. The interactions among the particles of the many-body system do not need to be small. The phenomenological theory of Fermi liquids was introduced by the Soviet physicist Lev Davidovich Landau in 1956, and later developed by Alexei Abrikosov and Isaak Khalatnikov using diagrammatic perturbation theory. The theory explains why some of the properties of an interacting fermion system are very similar to those of the ideal Fermi gas (i.e. non-interacting fermions), and why other properties differ. Important examples of where Fermi liquid theory has been successfully applied are most notably electrons in most metals and liquid helium-3. Liquid helium-3 is a Fermi liquid at low temperatures (but not low enough to be in its superfluid phase). Helium-3 is an isotope of helium, with 2 protons, 1 neu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
